System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Average Error: 0.1 → 0.1

Time: 4.7s

Precision: 64

Math

\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

↓

\[\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \left(1 - z\right) + \log \left(\sqrt{z}\right), y \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{z}}\right)\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right) \cdot y\right)\]

C

double f(double x, double y, double z) {
        double r248138 = x;
        double r248139 = 0.5;
        double r248140 = r248138 * r248139;
        double r248141 = y;
        double r248142 = 1.0;
        double r248143 = z;
        double r248144 = r248142 - r248143;
        double r248145 = log(r248143);
        double r248146 = r248144 + r248145;
        double r248147 = r248141 * r248146;
        double r248148 = r248140 + r248147;
        return r248148;
}

↓

double f(double x, double y, double z) {
        double r248149 = x;
        double r248150 = 0.5;
        double r248151 = y;
        double r248152 = 1.0;
        double r248153 = z;
        double r248154 = r248152 - r248153;
        double r248155 = sqrt(r248153);
        double r248156 = log(r248155);
        double r248157 = r248154 + r248156;
        double r248158 = 2.0;
        double r248159 = cbrt(r248155);
        double r248160 = log(r248159);
        double r248161 = r248158 * r248160;
        double r248162 = r248151 * r248161;
        double r248163 = fma(r248151, r248157, r248162);
        double r248164 = r248160 * r248151;
        double r248165 = r248163 + r248164;
        double r248166 = fma(r248149, r248150, r248165);
        return r248166;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

1. Initial program 0.1

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\]
3. Using strategy rm

4. Applied distribute-lft-in 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{y \cdot \left(1 - z\right) + y \cdot \log z}\right)\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(1 - z\right) + y \cdot \log \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right)\]

7. Applied log-prod 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(1 - z\right) + y \cdot \color{blue}{\left(\log \left(\sqrt{z}\right) + \log \left(\sqrt{z}\right)\right)}\right)\]

8. Applied distribute-lft-in 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(1 - z\right) + \color{blue}{\left(y \cdot \log \left(\sqrt{z}\right) + y \cdot \log \left(\sqrt{z}\right)\right)}\right)\]

9. Applied associate-+r+ 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log \left(\sqrt{z}\right)\right) + y \cdot \log \left(\sqrt{z}\right)}\right)\]

10. Simplified 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right)} + y \cdot \log \left(\sqrt{z}\right)\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + y \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) \cdot \sqrt[3]{\sqrt{z}}\right)}\right)\]

13. Applied log-prod 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + y \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right)}\right)\]

14. Applied distribute-rgt-in 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) \cdot y + \log \left(\sqrt[3]{\sqrt{z}}\right) \cdot y\right)}\right)\]

15. Applied associate-+r+ 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\left(\mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) \cdot y\right) + \log \left(\sqrt[3]{\sqrt{z}}\right) \cdot y}\right)\]

16. Simplified 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\mathsf{fma}\left(y, \left(1 - z\right) + \log \left(\sqrt{z}\right), y \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{z}}\right)\right)\right)} + \log \left(\sqrt[3]{\sqrt{z}}\right) \cdot y\right)\]

17. Final simplification 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \left(1 - z\right) + \log \left(\sqrt{z}\right), y \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{z}}\right)\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right) \cdot y\right)\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))